An arithmetic group associated with a Pisot unit and its symbolic-dynamical representatiom

To a given Pisot unit $\beta$ we associate a finite abelian group whose size appears to be equal to the discriminant of $\beta$. We call it the Pisot group and find its representation in the two-sided $\beta$-compactum in the case of $\beta$ satisfying the relation $Fin(\beta)=\Bbb Z[\beta]\cap[0,1)$. As a motivation for the definition, we show that the Pisot group is the kernel of some important arithmetic coding of the toral automorphism given by the companion matrix naturally associated with $\beta$.